I was about to say that I didn't think it likely that Dobby and Kreacher would become mortal enemies, however I just realized that the last person to talk to Kreacher that we know about was Dumbledore. So the question is, what did Dumbledore do with him?
He couldn't send him off to the Malfoys, Draco's mother being one of the last relatives, because of what he knows (though we can assume he has interacted with her in the past)
There's always Tonks, who is a cousin or something of Sirius', but I dont think she will want him. (would you?) I dont think Harry would get him for two reasons, not the least being that Harry despises him now that he knows he had a hand in Sirius' death.
And Harry isn't really entitle to Kreacher as he is not part of the Black family (and we dont yet know what will happen to Sirius's possessions)
So, maybe Dumbledore will take him to Hogwarts?

If he does, I am sure trouble will ensue between Dobby, who worships Harry (and loves his friends Hermione (who Kreacher calls a mudblood) and Ron(a traitor of pure bloods to Kreacher)), and Kreacher, who maliciously betrayed his own master because he held with the ideals of wizard purists.
It would be very interesting to see what would come out of their differences. Especially if Kreacher continues to spy within Hogwarts for Draco's family. Maybe there differences will be responsible for inciting a uprising amongst house-elves, forcing them to choose sides?